This is of course well known. The following observation is equally straightforward: we can apply Prop. REF  to any bipartite state by considering purifications. What is more, a version of the Schrödinger-HJW theorem assures independence of the choice of purification [1]}, [2]} (see also [3]}).
The PPT criterion thus becomes necessary and sufficient with respect to purifications. Of course, for pure states there are easier ways to check whether a state is entangled or separable.For instance, note that we have used the Schmidt rank in Prop.REF . Nevertheless, as we will see below the criterion works because it works on the level of purifications. This is best expressed in terms of channels.
